Notation Definition and 1000 Threads

Homework Statement I am trying to prove $$\vec{\nabla}(\vec{a}.\vec{b}) = (\vec{a}.\vec{\nabla})\vec{b} + (\vec{b}.\vec{\nabla})\vec{a} + \vec{b}\times\vec{\nabla}\times\vec{a} + \vec{a}\times\vec{\nabla}\times\vec{b}.$$ I can go from RHS to LHS by writng...

Homework Statement Let ##g \in S_n## and suppose that we know the cycle notation for ##g##. How can one compute its order without repeatedly composing ##g## with itself? Homework EquationsThe Attempt at a Solution [/B] I took the group ##S_4## and tried formulating a conjecture. I composed...

I was trying to understand electronic excitation in Xenon. Most of the research journals are denoting different excitation levels in terms of Paschen notation, which i think, is generally used to represent excited states of inert gases. Can someone please explain how to use this paschen notation...

We have this stationary metric, $$ds^2 = e^{2U}(dt+\omega_idx^i)^2 -e^{-2U}dx^2$$ The book wrote down the spin connections of this: $$ \omega^{0i}=\partial_ie^{U}e^0 +e^{3U}\partial_{[_i\omega _k]}e^k $$ and $$ \omega^{ij}= e^{3U}(\partial_{[_i\omega _j]}e^0-\partial_{[_ie^{-2U}\delta_j]k} )$$...

Two questions, both about adjacency matrices (graphs). The first, specific, the second, general. [1] I read: "Consider a directed graph and a positive integer k. Then the number of directed walks from node i to node j of length k is the entry on row i and column j of the matrix Ak..." [where A...

Having trouble understanding some notation: x(.) or C(.) (the dot is in fact in the center i.e mid-way vertically in the brackets). I have never run into this before? What does it mean if x is a function of "."?

The symbol for a Lorentz transformation is sometimes Λαβ and sometimes Λμν. When Λαβ is shown by a matrix, the upper index denotes the entries in the rows and the lower index denotes the entries in the columns. When Λμν is shown by a matrix, does the upper index still denotes the entries in...

Hi Everyone! I'm looking to prove $\nabla\cdot\left(\phi\textbf{u}\right)=\phi\nabla\cdot\textbf{u} + \textbf{u}\cdot\nabla\phi$ in index notation where u is a vector and phi is a scalar field. I'm unsure how to represent phi in index notation. For instance, is the first line like...

The context is Tonelli/Fubini. What's the difference between saying ## y\mapsto\int_Ef^y(x)dx ## is measurable and ## \int_Ef^y(x)dx ## is measurable

So I have seen time dilation written as all three of the following: t=τγ, Δt=Δτγ, dt=dτγ. I'm assuming this not to be the case, but just wanted to clarify that the third (differential) notation does not imply that t=∫τγ? That really wouldn't make sense (to me at least), so I'm assuming that...

So far the explanation on BJT working principle always explained in electron flow perspective. I felt it's hard to relate with conventional current direction notation when the explanation explained in electron flow direction notation. For example in NPN transistor, the C-E current flow made me...

Why is it that, ## \frac{a+\mathcal{O}(h^2)}{b+\mathcal{O}(h^2)} = \frac{a}{b}+\mathcal{O}(h^2) ## as ##h\rightarrow 0##? It seems like the ##\mathcal{O}(h^2)## term should become ##\mathcal{O}(1)##.

In GR an important, purely geometric equation is called Raychoudhuri's equation governing the behaviour of geodesic congruences which states that $$\frac{d\theta}{d\tau} = - \frac{1}3 \theta^2 - \sigma^{ab}\sigma_{ab} + \omega^{ab}\omega_{ab} -R_{ab} u^a u^a$$ where ##R_{ab}## is the ricci...

I apologize if this is in the wrong forum, but I'm not quite sure there's a category where this would fit in. Anyway, I'm doing an independent research/analysis of the RGB color space, and as I'm writing down my formulas and algorithms, I want to express in the math that certain variables are...

Hello everyone, The problem I am working on asks me to write a Cayley table for ##D_3## using cycle notation. However, I am having difficult with determining how to operate on a pair of "cycles." Here is my work so far: ##R_0 = e = ()## ##R_{120} = (123)## ##R_{240} = (312)## ##F_T =...

Classical Theory of Particles and Fields, by Boris Kosyakov, has the following in appendix A: (I don't own a copy of the book. This came to my attention through this physics.SE question, and it turned out that this part of the book is accessible through Amazon's peephole.) This completely...

I am refreshing my understanding of homology theory (well, recreating from scratch really!) after a thirty year break and there's something that bugs me in how the texts I've seen write about relative homology. The relative homology module ##H_q(X,A)## is defined as ##ker\ \partial_q^A/Im\...

Alright, so I am trying to make sense of this notation: From one source I have the transition rates notated such as Transition: 2-1, 3-1, 3-2, etc and the source to which I want to compare uses the notation Transition: 3P0-3P1, 3P0-3P2, 3P0-1D2, etc. How are these related? How can I tell...

Hi there, I am reading an article about the evolution of states. It is said that the system is initially in state |\Psi\rangle= a|a\rangle + b|b\rangle but after evolution, it becomes |\Psi'\rangle = a|a\rangle|c\rangle + b|b\rangle|d\rangle what is that (double ket) notation...

Below is an image from a website article discussing a Turing machine. It is supposed to represent a Turing machine, but I don't really follow the notation. I assume this is part of Wolfram's book, "New Kind of Science", which I do not have a copy of. I have only ever seen a Turing machine...

In a problem involving moments where we have a 2 point is space: m and n. We want the moment about M due to a force applied at N. We know the equation is Moment = r (m/n) * F So, does r (m/n) represent the (x,y,z) coordinates from the point N to M or M to N?

Homework Statement Show: εlmn detA = εijkAilAjmAkn Homework Equations not sure The Attempt at a Solution I'm really not sure where to begin. I understand index notation (at least I think I do). Can anyone help me get started? I think that once I have the first step, I'll be...

I'm confused over two different types of integral notation 1) ∫ (expression) dx and 2) ∫dx (expression) Are these the same thing? Example: Do ∫(x^2)dx and ∫dx (x^2) mean the same thing? Or is there a difference?

Hi, Can someone explain the difference between, say, \Lambda_\nu^\mu, {\Lambda_\nu}^\mu and {\Lambda^\mu}_\nu (i.e. the positioning of the contravariant and covariant indices)? I have found...

I'm taking a course in continuum mechanics this semester and the instructor is using a set of notes to teach out of it, problem is, I don't really like them. Can anyone recommend an engineering/applied physics oriented introductory continuum mechanics textbook that uses the Einstein summation...

Homework Statement This is my first exposure to Einstein notation and I'm not sure if I'm understanding it entirely. Also I added this class after my instructor had already lectured about the topic and largely had to teach myself, so I ask for your patience in advance... The question is...

Hi. Currently I'm taking an advanced particle physics course, and apparently Einstein notation takes up a lot in this course. Unfortunately for me, and several others in this course, we have never had anything with this kind of notation before. And pretty much from day one, we were put right...

I'm asked to prove the following using Levi-Civita/index notation: (\mathbf{a \times b} )\mathbf{\times} (\mathbf{c}\times \mathbf{d}) = [\mathbf{a,\ b, \ d}] \mathbf c - [\mathbf{a,\ b, \ c}] \mathbf d \ I'm able to prove it using triple product identities, but I'm completely stuck...

Homework Statement A.) Show that \epsilon_{ijk}A_{k,j} represents the curl of vector A_k B.) Write the expression in indicial nottation: \triangledown \cdot \triangledown \times A 2. The attempt at a solution I'm hoping that if I can get help on part A.) it will shed light on...

First day of school and I don't have much work. So I got bored and read ahead a bit. From a textbook my prof wrote: Can someone explain why they used the "$\equiv$"? I think it means "equivalent", but I'm not sure, but when are times you want to use that symbol rather than "equals"? What's the...

If you research the materials semiconductors are made of you will find things such as aluminum gallium arsenide. How do you determine what the p-type and n-type are made of? Shouldn't there be a structural difference in order to create band gap?

I don't know if I've ever encountered a differential term with a modulus around it (or if I have, ignored it). Here's an example: $ \int\limits_{\gamma}{\rho(z)}{|{dz}|} $ If it was simply $ \int\limits_{\gamma}{|{dz}|} $ I imagine this is the length of the curve $\gamma $, but what might the...

Sorry for a long post. I am looking for a clear and concise way to explain how to compute coordinates when changes of basis or linear operators are involved. I would like to avoid the summation notation as much as possible and use the definition of matrix multiplication only in the beginning...

I am reading Berrick and Keating's book on rings and modules. Berrick and Keating indicate a right module over a ring R as M_R, but in a left module the subscript is to the left ... can someone help me with the Latex script to achieve this? An example of "left-subscript" notation appears in...

Homework Statement What's the domain of 14/x2-x-6, in interval notation? Homework Equations \underline{14}x2-x-6 The Attempt at a Solution [3, + infinity) [-2,- infinity) Sorry, I can't find the infinity symbol Thx

First off: I think I understand the chain rule and how it derives from \lim_{h \to 0} \frac{ f(x+h)-f(x)}{h} and how to apply the chain rule when taking the derivative of an implicit function. The textbook I am reading Applied Calculus (by B. Rockett) uses the following example on...

Homework Statement A current element is 3.0mm long, centerd on the origin of cartesian coordinates, and carries a current of 2.5A in the direction ez. What field does it produce at the point (3.0, 0, 4.0) Homework Equations This is obviously a Lorentz force law question so . . . ∂B =...

I'm looking through my lecture notes, (studying relativistic corrections/perturbation theory using hydrogen), and I seem to have a mind block with one of the equations (the last one from the 3 in the middle). I know that the kinetic energy and coulomb potential has been subbed in for the...

Hi! For the following functions, what are their big-O notation? 1. n^(n-1) 2. (n-1)^n Should their big-O notations be the same as the original functions? i.e. 1. O(n^(n-1)) = n^(n-1)? 2. O((n-1)^n) = (n-1)^n? Please help! Many thanks!

I am trying to reconcile the following statement: " ||u||<=eps*||f|| means ||u||=O(eps) ("||u|| is order eps")... " with the limit definition of "big O"; considering it's not clear that ||u|| here even depends on eps: " lim as eps goes to 0 of ||u||/eps, by definition of "big O", should...

Hi, I'm teaching myself tensor analysis and am worried about a notational device I can't find any explanation of (I'm primarily using the Jeevanjee and Renteln texts). Given that the contravariant/covariant indices of a (1,1) tensor correspond to the row/column indices of its matrix...

I want to show, in a talk, how to build the Predator-Prey (Lotka Volterra) model. It starts off with the "Population Law of Mass Action" i.e. "At time t the rate of change of one population due to interaction with another is proportional to the product of the two populations at that time t"...

Hi guys, I m reading some theoretical physics paper that requires knowledge of dirac notation if someone could point me out to a good tutorial on it I come from a math background but I am studying this paper with my supervisor.

Hi, I have a question about : http://imgur.com/RU7PvtJ I actually understand what I need to do. I need to see if both one sided limits are the same to establish that the limit exists. The only thing which I just find weird is the "since y --> 2^(-) implies y<-2" Can somebody explain me...

I've been reading about algebraic geometry lately. I see that a lot of authors use ##V^\vee## to denote the dual space of a vector space ##V##. Is there any particular reason for this? The only reason I could think of is that this notation leaves us free to use ##R^*## to denote the units of...

Hi. In this development (c ∇+ d A)(c ∇+dA)= c^{2} ∇^{2} + d^{2}A^{2} + cd A∇ + cd ∇A (c ∇+ d A)^{2}= c^{2} ∇^{2} + d^{2}A^{2} + cd A∇+ cd A∇+ cd (∇A) I feel like we have "two" different ∇ operators. At the end of the first line ∇ acts on A and the test function (not shown). At the...

Given vector spaces V, W, and a function T:V→W , state the two equations that the function T must satisfy to be a linear function. Does T:V→W mean a function that maps vectors in V into W? Or what does this actually mean?

Homework Statement This isn't really a problem so much as me not being able to see how a proof has proceeded. I've only just today learned about Dirac notation so I'm not too good at actually working with it. The proof in the book is: |Z> = |V> - <W|V>/|W|^2|W> <Z|Z> = <V - ( <W|V>/|W|^2 ) W|...

Hello guys, Let ##T: \mathbb{R^2} \to \mathbb{R^2}##. Suppose I have standard basis ##B = \{u_1, u_2\}## and another basis ##B^{\prime} = \{v_1, v_2\}## The linear transformation is described say as such ##T(v_1) = v_1 + v_2, T(v_2) = v_1## If I want to write the matrix representing ##T##...

Homework Statement Homework Equations The Attempt at a Solution Hello, I am having some confusion over the notation used in matlab. I don't really know what they mean A = [1:3; 4:6; 7:9] A = 1 2 3 4 5 6 7 8 9 A(1:2, 1:2) ans = 1 2...